On the residual lifelengths of the remaining components in an n−k+1n−k+1 out of nn system

Suppose that a system consists of nn independent components and that the lifelength of the iith component is a random variable Xi(i=1,2,…,n). For k∈{1,2,…,n−1}k∈{1,2,…,n−1}, denote by X1(k),X2(k),…,Xn−k(k), the residual lifelengths of the remaining functioning components following the kkth failure in the system. We discuss the joint distribution of these exchangeable random variables. In addition, we identify the conditions sufficient to guarantee the independence of the residual lifelengths.